ANFIS with correlated residuals

ANFIS refers to 'adaptive network-based fuzzy inference system' which fits a Sugeno-type fuzzy model to a training dataset.

This Python project anfis_co deals with the situation when there is a certain degree of correlation between the residuals in an 'Output = ANFIS(Input) + Residuals' model. For example, under many conditions, the residuals may have spatial or temporal correlation.

The regular anfis package in Python can be found at https://github.com/twmeggs/anfis, from whch this project was modified.

Installation

Requires Python >= 3.0. (All the modules were tested in Python 3.5.2.)

This project was not published as a package in PyPI and cannot be installed by running $ pip install. Therefore, in order to use the package, you need to download the files and use them as a self-written package.

anfis_co depends on:

• numpy
• sklearn
• scikit-fuzzy
• matplotlib

Usage

• anfis_co.py

  Input: {x: independent variables in training data, y: response in training data, C: covariance matrix of residuals at different locations or time}.

  Output: parameters of fuzzy inference system fitted to training data {x, y, C}.

• test_spatial.py

  In most conditions, covariance matrix C is unknown and need to be estimated. Therefore, an iterative method need to be used. test_spatial gives an example showing how to conduct an iterative algorithm for (a) spatial kernal estimation using sklearn.gaussian_process, and (b) fuzzy inference system estimation using anfis_co.

Notes

• Check generalized least squares in Wikipedia to see how the correlation in residuals affecting the inference of model parameters.
• Make C = I if no correlation exists in residuals. In this case, anfis_co will produce the same results with anfis in original anfis package at https://github.com/twmeggs/anfis.